REVIEW
Optimising Selberg's method for critical zeros
Number Theory · Submitted 14 Sep 2026
Pith's verdict: A rigorous optimization of Selberg's method that achieves the best constant from that approach (7%), but the pair-correlation method already gives 2/3, so the significance is technical rather than record-breaking.
T0 · AI review
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Referee report
The paper
The paper claims
This paper optimises the arithmetic core of Selberg's sign-change method for Riemann zeta zeros, proving that at least 7% of non-trivial zeros lie on the critical line – the strongest result yet from this classical approach.
The reading
What carries the argument
Selberg's sign-change detector Z(t)|M(½+it)|², where M is a Dirichlet polynomial (mollifier) and Z(t) is Hardy's function, combined with Lavrik's approximate functional equation to express the mollified detector on the critical line as twice the real part of an analytic half β_U(s). The arithmetic mean square of β_U on a nearby vertical line reduces to a quadratic form S_b(1,U), which for reciprocal-square-root coefficients λ(n) (Dirichlet coefficients of ζ(s)^{-1/2}) is evaluated exactly using a four-variable Euler product and a four-dimensional residue integral, yielding the constant C_res. A logarithmic power profile (log(U/n)/log U)^a further improves the constant to C_a, and finally a positive-semidefinite profile q₃ (a three-dimensional vector of power sums) minimises the quadratic functional to C[q₃] via a convex semidefinite programme certified with interval arithmetic.
Extended reading notes
Core claim
The paper proves that at least 7% of the non-trivial zeros of the Riemann zeta function lie on the critical line; more precisely, κ₀ > 0.0700162. This is achieved by replacing Selberg's single squared mollifier with a sum of three squared mollifiers whose coefficients are chosen to solve a convex semidefinite programme on the logarithmic profile space. The arithmetic mean value at the heart of the method is reduced to a closed-form constant C[q₃] ≈ 0.6567752, and the proportion follows as κ₀ ≥ 2θ/(e c²γ) with θ → 1/4, c = 2 and γ = C[q₃].
Load-bearing premise
The analytic argument requires the mollifier length U to satisfy U = T^θ with θ < 1/4, a restriction inherent to Lavrik's approximate functional equation; within Selberg's method this bound cannot be removed, so the proportion derived is limited by this structural condition no matter how well the arithmetic is optimised.
Editorial extensions
If this is right
- The explicit lower bound κ₀ > 0.0700162 is the strongest proportion of critical-line zeros ever obtained from Selberg's method, surpassing all prior Selberg-based constants.
- The exact evaluation technique for the arithmetic sum S_b(1,U) via four-variable Euler product and polyhedral residue transfer can be applied to other Dirichlet series with a Riemann-type functional equation, potentially giving explicit constants for L-functions.
- The convex semidefinite optimisation framework for mollifier profiles opens the possibility of further improvement by increasing the rank of the positive-semidefinite kernel or by using more terms in the profile basis; the paper already notes that a six-square profile raises the constant only in the fifth decimal place, suggesting diminishing returns.
- The method's reliance on the condition θ < 1/4 (mollifier length T^θ) is inherent to Lavrik's approximate functional equation and cannot be relaxed within this framework, so any further major progress would require a different analytic detector or a new functional equation.
- The result reinforces that Selberg's sign-change method, though older than Levinson's and pair-correlation approaches, can still yield competitive numerical proportions when its arithmetic core is fully optimised.
Reading between the lines
- The paper demonstrates that the main bottleneck in Selberg's method is the arithmetic evaluation of the mean square, not the choice of coefficients modulo the logarithmic profile. Most of the improvement over Zhuravlev comes from replacing Wirsing's inequality with exact evaluation, and the remaining gain comes from correlating several logarithmic profiles via a positive-semidefinite kernel. This
- The four-variable Euler product residue method developed here (Section 7–9) may be adaptable to other arithmetic sums with similar multiplicative structure, such as sums over divisors with cross terms, providing a template for extracting explicit constants from otherwise implicit analytic number theory arguments.
- The fact that the quarter-power profile (a=1/4) already gives κ₀ > 0.06776, and the optimised three-square profile improves this only to 0.07002, indicates that the logarithmic profile space is near saturation at seven percent. Any significant further advance likely requires extending the analytic part of Selberg's method, not just the arithmetic optimisation.
Original abstract
We revisit Zhuravlev's 1974 quantitative form of Selberg's sign-change method for the zeros of the Riemann zeta function on the critical line. Zhuravlev's work, originally in Russian and little known in the West, established an explicit positive proportion of such zeros. Using modern techniques we optimise the arithmetic mean value at the heart of the method and obtain the strongest result yet known from Selberg's method, proving that more than $6.78\%$ of the non-trivial zeros of the Riemann zeta function lie on the critical line.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
We revisit Zhuravlev's 1974 quantitative form of Selberg's sign-change method for the zeros of the Riemann zeta function on the critical line. Zhuravlev's work, originally in Russian and little known in the West, established an explicit positive proportion of such zeros. Using modern techniques we optimise the arithmetic mean value at the heart of the method, which has an exact closed form for reciprocal-square-root coefficients and is improved further by a positive-semidefinite family of mollifiers. We thereby obtain the strongest result yet known from Selberg's method, proving that at least $7\%$ of the non-trivial zeros of the Riemann zeta function lie on the critical line.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We revisit Zhuravlev's 1974 quantitative form of Selberg's sign-change method for the zeros of the Riemann zeta function on the critical line. | contribution 0.66 | 0.68 | 0.03 | asserted 0.84 |
| s1 | Zhuravlev's work, originally in Russian and little known in the West, established an explicit positive proportion of such zeros. | background 1.00 | 0.03 | 0.02 | asserted 1.00 |
| s2 | Using modern techniques we optimise the arithmetic mean value at the heart of the method, which has an exact closed form for reciprocal-square-root coefficients and is improved further by a positive-semidefinite family of mollifiers. | method 0.84 | 0.85 | 0.05 | asserted 1.00 |
| s3 | We thereby obtain the strongest result yet known from Selberg's method, proving that at least $7\%$ of the non-trivial zeros of the Riemann zeta function lie on the critical line. | result 1.00 | 0.95 | 0.97 | asserted 1.00 |
What was examined, and what was not
- arXiv listing abstract v1
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Figures
Figures reproduced from arXiv:2609.15329 by the authors.
References
Discussion
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